The liftability of elliptic surfaces [Elektronische Ressource] / vorgelegt von Holger Partsch

Holger Meixner

Sujets

• Mathematics

Informations

• Publié par: heinrich-heine-universitat_dusseldorf
• Publié le: 01 janvier 2011
• Nombre de lectures: 20
• Langue: Deutsch

Extrait

The Liftability of Elliptic Surfaces

Inaugural-Dissertation

zur Erlangung des Doktorgrades derMathematisch-NaturwissenschaftlichenFakulta¨t derHeinrich-Heine-Universit¨atDu¨sseldorf

vorgelegt von

Holger Partsch aus Augsburg

D¨usseldorf,Dezember2010

Aus dem Mathematischen Institut derHeinrich-Heine-Universit¨atD¨usseldorf

Gedruckt mit der Genehmigung der Mathematisch-NaturwissenschaftlichenFakult¨atder Heinrich-Heine-Universit¨atDu¨sseldorf

Referent:Prof.Dr.StefanSchr¨oer

Koreferent: Dr. habil. Cristiana Bertolin

Tagderm¨undlichenPru¨fung:21.01.2011

Erstellungsdatum der elektronischen Version: 25.01.2011

Contents

Acknowledgments v Zusammenfassung vii Summary ix Introduction xi Background and motivation xiv Notations and conventions xvi Chapter 1. Smooth elliptic fibrations and elliptic fibre bundles 1 1. Preliminaries: Deformation theory and thickened schemes 1 2. Elliptic fibre bundles: Definition and examples 3 3. Elliptic curves and their moduli 4 4. Deformations of Jacobian elliptic fibre bundles 5 5. Non-liftable elliptic fibre bundles 8 6. Deformations of elliptic torsors 9 7. Elliptic fibre bundles of Kodaira dimension one 12 8. Bielliptic surfaces 15 Chapter 2. Liftings of semistable elliptic fibrations 27 1. Semistable genus-1 curves and generalized elliptic curves 27 2. Modular invariants and isomorphism types over complete bases 29 3. Moduli of generalized elliptic curves with level structures 31 4. Level structures and liftability 32 5. Liftings of generalized elliptic curves with separable modular invariants 34 6. The constructions 38 7. Liftings of principal homegenous spaces 44 Chapter 3. Liftability under tameness assumptions 49 1. Tame coverings 49 2. Tame Jacobian fibrations 50 3. Non-Jacobian fibrations of period prime top56 4. Simultaneous desingularization 57 Bibliography 61

iii

Acknowledgments

Writing this thesis has been challenging in many ways. Now, looking back I want express my gratitude to those without whom this work would not exist in its current form. Firstofall,IwanttothankmyadvisorStefanSchro¨erforacquaintingme with the fascinating topic of this work and for his substantial and inspiring support during its origination. Without my colleagues Philipp Gross, Sasa Novakovich andFelixSchu¨llerlifeatD¨usseldorfwouldhavebeenbothlessenjoyableandless productive. Especially, I want to thank Philipp for his friendship and his willingness to share his profound knowledge on abstract algebraic geometry. I also want to thankUlrikeAlba,PetraSimonsandJuttaGonskafromthestaffinDu¨sseldorf and Essen for their constant support and patients in the last three years. I am also in dept to Christian Liedtke “the best assistant of all times”, who alwaysfoundthetimeforhelpfuldiscussions.BeyondDu¨sseldorfIwanttothank WilliamLang,HiroyukiItoandMathiasSchu¨ttforhelpfuldiscussionandcomments and for reading parts of this work. The concentrated brainpower atmathoverflow and especially BCrnd deserve to be mentioned here. I also want to thank Duco van Straten who made it possible for me to be supported by the SFB45. MyfriendsinBonn,Du¨sseldorfandEssendeservetobethankedforexcusing so many times all the different forms of research inflicted tieredness. A special thanksgoestoAndreArtmann,AnnaFludder,RolandH¨utzen,KatharinaJacobi and JUM Lange for hearing my out on the lesser problems that appear on the boarder line between life and math. Towards my family I will refrain from using words for things which are funda- mental in their goodness and which I have taken for garanted so many years now. Among the former I especially place Rolf Loosen with his entire being.

v

Zusammenfassung

DasThemadieserArbeitistdieLiftbarkeitelliptischerFl¨achen.EinSchemaX u¨bereinereinemK¨orperk⊃F p heißt liftbar nach Characteristik 0 wenn ein lokaler Ring (R,m) existiert mitRm≃kundZ⊂R, sowie ein flaches SchemaX¨uberR mitX ⊗ R k≃Xlltetiiph,scnnweniE.¨lFeehcaßiehsctiiplleeineesignuresaFeh besitzt, d.h. einen Morphismus auf eine Kurve, dessen generische Faser eine glatte ¨ KurvevomGeschlecht1ist.ElliptischeFla¨chenexistierenimUberfluss,dennviele Fla¨chevonKodaira-Dimensionkleinergleich0sindelliptisch,undjedeFl¨achevon Kodaira-Dimension 1 besitzt eine elliptische oder quasi-elliptische Faserung. Zur Untersuchung der Liftbarkeit elliptischer Faserungen ziehen wir die Mod- ultheorie elliptischer Kurven heran. Im ersten Kapitel der vorliegenden Arbeit studieren wir Deformationen und Liftungen glatter elliptischer Faserungen. Es stellt sich heraus, dass die Deformationstheorie dieser Objekte so gut kontrollier- bar ist, dass wir in der Lage sind, nicht liftende Beispiele elliptische Faserungen u¨berKo¨rpernderCharacterisik2und3zukonstruieren.Eshandeltsichumdie ersten bekannten Beispiele. Auch konstruieren wir eine Klasse elliptischer Faserun- genderenLiftbarkeit¨aquivalentzueineroffenenVermutungvonOortist.Das unterstreichtdieKomplexit¨atdesLiftungsproblemsf¨urelliptischeFaserungen.Als weitereAnwedungklassifizierenwirdieDeformationenbielliptischerFla¨chenund zeigen deren Liftbarkeit. ImzweitenKapitelbesch¨aftigenwirunsmitsemistabilenelliptischenFaserun- gen.MittelsderModultheorief¨urverallgemeinerteelliptischeKurven,entwickelt vonDeligneundRapoportunderweitertdurchConrad,k¨onnenwirzeigen,dass jede semistabile elliptische Faserung mit Schnitt und separabler modularer Invari- ante nach Characteristik 0 lifted. Das dritte Kapitel handelt von elliptischen Faserungen die bestimmte Zahmheits- eigenschaftenerfu¨llen.NachAusschlussvonCharakterisik2und3zeigenwir,dass jedeJacobischeelliptischeFaserungmitzahmermodularerInvariantelifted.F¨ur nicht jacobische Faserungen gilt ein vergleichbares Resultat, falls ein Multischnitt vom Grade prim zupexistiert. Als Fazit erhalten wir, dass obgleich nicht liftende elliptische Faserungen exis- tieren, dieses Verhalten nicht das typische ist. Istpssneegiwshrsuaz¨mtileiVn arithmetischen Invarianten der fraglichen Faserung hinreichend groß, so gilt Lift- barkeit.

vii

Summary

The topic of this work is the liftability of elliptic surfaces. A schemeXover a fieldk⊃F p is calledliftable to characteristic zero, if there is a local ring (R,m) withRm≃kandZ⊂R, as well as a flat schemeXoverR, such thatX ⊗ R k≃X. A surface is calledelliptic morphism to a curve,if it has an elliptic fibration, i.e. a such that the generic fibre is a smooth genus-1 curve. Elliptic surfaces exist in abundance because they are common in Kodaira dimension less than one and every surface in Kodaira dimension one has a unique elliptic or quasi-elliptic fibration, given by the canonical bundle. To investigate the liftability of elliptic fibrations, we make extensive use of the moduli theory of elliptic curves. In the first Chapter of this work, we study deformations and liftings of smooth elliptic fibrations. It turns out that we can control their deformations fairly well, which allowes us to give examples of non- liftable elliptic fibrations of Kodaira dimension one over fields of characteristic two and three. Those are the first examples currently known. We also construct a class of elliptic surfaces whose liftability is equivalent to an open conjecture of Oort. This illustrates the complexity of the lifting problem for elliptic surfaces. To give a further application, we classify deformations of bielliptic surfaces and show that they are liftable. In the second Chapter we are concerned with semistable elliptic fibrations. Using the moduli theory of generalized elliptic curves developed by Deligne and Rapoport and extended by Conrad, we can show that every semistable elliptic fibration, possesing a section and having a separable modular invariant, is liftable to characterstic zero. The third chapter deals with elliptic fibrations satisfying certain tameness prop- erties. Excluding characteristic two and three, we prove that a Jacobian elliptic fi- bration with tame modular invariant is liftable in the category of algebraic spaces. For non-Jacobian fibrations we have a similar result, given the existence of a mul- tisection of degree prime top. As a conclusion, we can say that, although non-liftable elliptic fibrations do exist, this is not the typical behaviour. Given thatpis sufficiently large in compar- ison to certain arithmetic invariants of the surface in question, liftability is seen to hold true.

ix

Introduction Drei Orangen, zwei Zitronen:- Bald nicht mehr verborgene Gleichung, Formeln, die die Luft bewohnen, AlgebraderreifenFr¨uchte! 1 The question of liftability arises naturally in the modern formulation of alge- braic geometry due to Grothendieck, because algebraic geometry can be done not just over fields, but over more general bases such as arbitrary rings. Now letRbe a local ring with residue fieldk. Given a schemeXoverk, we can ask for alifting ofXoverR, this is aflatR-schemeX, withX ⊗ R k≃X. In other words, a lifting ofXis a family overRwhich contains the givenX as a special fibre. If the ringRis ak-algebra, this question is trivial, because we can always take a product family. However, the question becomes non-trivial, if Ris a ring of mixed characteristic, sayingkis of positive characteristic and the fraction field ofR In that case we speak of a lifting ofis of characteristic zero.X to characteristic zero. We are going to study liftability for surfaces having a fibration onto a curve such that the generic fibre is a smooth curve of genus one. This question was posed by Katsura and Ueno in [KU85]. Our results are twofold: We construct non-liftable elliptic surfaces of Kodaira dimension one over fields of characteristic two and three, showing that liftability does not hold in general. For generalpwe construct a special class of elliptic fibrations, whose liftability would imply an open conjecture of F. Oort. This illustrates the complexity of the problem. On the other hand, we establish a series of affirmative lifting results for certain classes of elliptic surfaces. This work is organized in three Chapters. The overall principle of its organi- zation is derived from its mathematical objects of study: Beginning with the most special class of elliptic fibrations, namely the smooth ones, we work towards higher degrees of generality in the following Chapters. In thefirst Chapterwe study smooth elliptic fibrations. elliptic fibra- Smooth tionsoverproperbasescanbeinterpretedasellipticfibrebundles,i.e.´etalelocally over the base they become constant fibrations. We classify deformations of Jacobian elliptic fibre bundles (those possesing sections), and prove that an arbitrary elliptic fibre bundle has a formal lifting if and only if its Jacobian does so (Theorem 1.6.5). In the Kodaira dimension one case, we get: Theorem(1.7.1).IfXCis an elliptic fibre bundle of Kodaira dimension one, then its unique elliptic fibration extends in a unique way to every deformation. 1 This is the first stanza of the poem “Drei Orangen, zwei Zironen” by Karl Krolow. A word by word english translation reads: “Three oranges, two lemons:- / Equation, soon no longer concealed, / Formulas, inhabiting the air, / Algebra of ripe fruit!” xi